Solving Trigo Limit Type 0/0 using L'Hopital's Rule | Sin x - x / x - tan x

[Deathfish]

Homework Statement

Find limit of (sin x - x)/(x - tan x) as x approaches zero

Homework Equations

Type 0/0 , use L'Hopital's rule, differentiate.

The Attempt at a Solution

Every time I apply the rule it gets more complicated

(cos x - 1)/(1 - sec^2 x)
(sin x)/(2 sec^2 x tan x)
(cos x)/(2 Sec^4 [x] - 4 Sec^2 [x] Tan^2 [x])

etc etc please help

 

[micromass]

Homework Statement

Find limit of (sin x - x)/(x - tan x) as x approaches zero

Homework Equations

Type 0/0 , use L'Hopital's rule, differentiate.

The Attempt at a Solution

Every time I apply the rule it gets more complicated

(cos x - 1)/(1 - sec^2 x)
(sin x)/(2 sec^2 x tan x)

Stop here, write sec and tan in terms of sin and cos. Eliminate sin from numerator en denominator.

 

[Deathfish]

Ok usually how do you decide when to stop differentiation and use alternative method?

 

[micromass]

Ok usually how do you decide when to stop differentiation and use alternative method?

The idea is to simplify the formula enough after each differentiation. That way you can see that you need to stop differentiation. Here, you need to write everything in sin and cos. Then you'll see after the second differentiation that you can cancel thingies.